Funkcialaj Ekvacioj 18 (1975) 99-106

Prolongations and Prolongational Limit Sets inNon-Autonomous Differential Equations

By S. M. Shamim IMDADI and M. Rama Mohana RAO

(University of Kashmir and Indian Institute of Technology)

1. Quite recently, G. R. Sell [6, 7] has shown that there is a way of viewingthe solutions of nonautonomous differential equations as dynamical systems. Thisview point is very general and includes all differential equations satisfying onlythe weakest hypotheses. In [6], by introducing the concept of the “set of limit-ing equations” for a given differential equation, he has investigated some of theimplications of topological dynamics in this setting. Further, by using the con-

cept of $¥omega-$fimit set in [7], G. R. Sell has established the basic relationship bet-ween the solutions of a given differential equation and the solutions of the cor-responding limiting equations. His results generalize some of the earlier worksof L. Markus [4] and R. K. Miller [5].

In this paper we introduce concepts of prolongation and prolongationaf limitsets for the solution of a given differential equation in the usual way and studythe properties of these sets. These are then used to discuss the behavior ofsolutions of limiting equations and to discuss some of the stability properties ofthe solutions of the given differential equation. The above concepts are well-known in the theory of dynamical systems [1] and its usefulness in stabilitytheory have been established in recent years by many authors including Ura [8]and Bhatia and Hajek [2]. Section 2 deals with definitions and basic lemmas.In Section 3, we obtain some results which relate the behavior of a positivelycompact solution of the given differential equation with the behavior of solutionsof the corresponding limiting equations. This study includes some of the resultsof G. R. Sell [7]. In Section 4, we have some theorems which characterize thestability properties of solutions of a given differential equation in terms of itsprolongation and prolongational limit sets. To motivate the nature of the resultcontained in Theorem 4. 1 we recall the following well-known theorem of T.Ura for dynamical systems defined on locally compact spaces:

Theorem (T. Ura, [8]). A compact set $M$ is stable if and only if $D^{+}(M)$

$=M$.Further we establish some relationship between the stability of the local

dynamical system $¥pi$ (see [2] for definition) and the stability of solution of thecorresponding differential equation.

100 S. M. Shamim IMDADI and M. Rama Mohana RA0

2. Let $W$ be an open set in $R^{n}$ , euclidean $¥mathrm{r}¥mathrm{c}$-space. The euclidean norm on$R^{n}$ will be denoted by $|x|$ . Let $C=C(W¥times R, R^{n})$ denote the set of all conti-nuous functions $f$ defined on $W¥times R$ with values in $R^{n}$ . We shall say that afunction $f$ is admissible [6] if (i) $f¥in C$, and (ii) the solutions of the differen-tial equation $x^{¥prime}=f(x, t)$ are unique. By the second condition we mean thatgiven any point $(x_{0}, t_{0})$ in $W¥times R$ , there is precisely one solution $¥phi$ of $x^{¥prime}=f(x$ ,$t)$ that satisfies $¥phi(t_{0})=x_{0}$ . It is evident that if $f$ is an admissible function, thenevery translate $f_{¥tau}$ of $f$ (where $f_{¥tau}$ ($x$, $t)=f(x,$ $¥tau+t)$ ) is an admissible function.Also, if $f$ satisfies the global existence property, then so does each $f_{¥mathrm{r}}$ . Let $F$

$=¥{f_{¥tau} : ¥tau¥in R¥}$ be the space of translates of /, then $F$ is a subset of $C$. Now let$f$ be an admissible function and consider the space of translates $F$ in the com-

pact open topology [3] on $C$. Let $¥mathrm{F}^{*}=¥mathrm{C}1$ $F$ (that is, the closure in the compactopen topology) be the hull of $f$. We shall say that $f$ is regular [6] if everyfunction $f^{*}$ in the hull $F^{*}$ is admissible. It is shown in [6] that the mapping$¥pi^{*}:$ $F^{*}¥times R¥rightarrow F^{*}$ , defined by $¥pi^{*}(f^{*}, t)=f_{t^{*}}$, is a dynamical system on $F^{*}$ , when$p*$ has compact open topology. Let $¥pi_{f^{*}}$ denote the flow $¥pi^{*}(f, t)$ with initialpoint /. We shall say that the motion $¥pi_{f^{¥prime}}*$ is positively compact [6] if the clo-sure of $¥{¥pi^{*}(f, t) : t¥geq 0¥}$ lies in a compact subset of $F^{*}$ {Negative compactnessand compactness are defined similarly).

Definition of Limiting Equation [6]. Let $f¥in C$ and let $F^{*}$ be the hullof $f$ (Neither regularity nor admissibility of $f$ will be important here). Let $¥pi^{*}$

$(f_{ }^{*}., t)=f_{t^{*}}$ be the flow on $F^{*}$ and let $¥Omega_{f^{*}}$ denote the $¥omega$-limit set of $f$ in thisflow. If the $¥omega$ -limit set $¥Omega_{f^{*}}$ of $f$ in $F^{*}$ is nonempty, then we say that the setof limiting equations for

(2. 1) $x^{¥prime}=f(x, t)$

is the set of all differential equations of the form(2. 2) $x^{¥prime}=f^{*}(x, t)$

where $f^{*}¥in¥Omega_{f^{*}}$ .Standing Hypothesis : Throughout the paper we shall consider only those

differential equations (2. 1) for which $f$ does not belong to either the positivelimit set $l2_{f^{*}}$ or the negative limit set $A_{f^{*}}$ of the motion $¥pi_{f^{*}}$ . This is justifiedas our interest is to compare the behavior of solutions of (2. 1) with the corre-sponding behavior of solutions of (2. 2). This would imply that $F$ is an openset in $F^{*}$ , as is easily proved.

Let $f¥in C$ be an admissible function and let $¥phi(t)=¥phi(x, /, t)$ be the solutionof (2. t) that satisfies $¥phi(x, f, ¥mathrm{O})=x$ . Let $I_{(x’ f)}=(¥alpha, ¥beta)_{-}¥mathrm{b}¥mathrm{e}$ the maximal intervalof definition of ($¥rho$ . We shall say that the solution $¥phi$ is positively compact if theset $¥{¥phi(t) : 0¥leq t<¥beta¥}$ lies in a compact set in $W$ {Negative compactness and com-

pactness are defined similarly). Since $I_{(x’ f)}$ is maximal, it follows that $¥beta=+$

Prolongations $ar¥iota d$ Prolorigational Limit Sets 101

$¥infty$ , whenever $¥phi$ is positively compact. If $¥phi$ is compact, then $I_{(x’ f)}=R$ , thatis, $¥phi$ is defined for all $t$ in $R$ .

Let $f¥in C$ be a regular function. Then the mapping [6, Theorem 8]$¥pi$ $(x, f*; t)=(¥phi(x, f^{*}, t), f_{t}^{*})$

defines a local dynamical system on $W¥times F^{*}$ , where $F^{*}$ is the hull of /. It isclear that the motion $¥pi(x, f^{*} ; t)$ is defined for all $t¥geq 0$ if and only if the solu-tion $¥phi(x, f^{*}, t)$ is defined for all $t¥geq 0$ .

For $(x, f^{*})¥in W¥times F^{*}$ , $¥mathcal{T}(x, f^{*})=$ $¥{¥pi(x, f^{*} ; t) : t¥in R¥}$ is the trajectory through$(x, f^{*})$ . Similarly $¥uparrow¥wedge+(x, f^{*})=$ $¥{¥pi(x, f’*; t) : t¥geq 0¥}$ and $¥mathcal{T}^{-}(x, f^{*})=$ { $¥pi(x, f^{*}; t)$ :$t¥leq 0¥}$ are defined to be, respectively, the positive and negative semitrajectorythrough $(x, f^{*})$ .

The limit sets, prolongations, and prolongational limit sets may now be de-fined as follows.

Definition 2. 1. For each $(x, f^{*})=¥leftarrow W¥times F^{*}$ , let$¥Omega(x, f^{*})=$ { $(y,$ $g)¥in W¥times F^{*}:$ $¥pi(x,$ $f^{*}$ ; $t^{t})¥rightarrow(y$ , $g)$ for some sequence $ t^{f}¥rightarrow+¥infty$},$D^{+}(x, f^{*})=¥{(y, g)¥in W¥times F^{*}:$ $¥pi(x^{t}, f^{t}; t^{t})¥rightarrow(y, g)$ for some sequences $t^{f}¥geq 0$ and$(x^{t}, f^{t})¥in W¥times F^{*}$ such that $(x^{t}, f^{t})¥rightarrow(x, f^{*})¥}$ , and$J^{+}(x, f^{*})=¥{(y, g)¥in W¥times F^{*}:$ $¥pi(x^{t}, f^{t}; t^{t})¥rightarrow(y, g)$ for some sequences $ t^{f}¥rightarrow+¥infty$

and $(x^{t}, f^{t})¥in W¥times F^{*}$ such that $(x^{f}, f^{t})¥rightarrow(x, f^{*})¥}$ .$¥Omega(x, f^{*})$ is called the positve limit set, $D^{+}(x, f^{*})$ the positive profongation, and$J^{+}(x, f^{*})$ the positive profongationaf limit set of $(x, f^{*})$ . The negative limit set$A^{*}(x, f^{*})$ , the negative prolongation $D^{-}(x, f^{*})$ , and the negative prolongationallimit set $J^{-}(x, f^{*})$ are defined similarly.

Now consider the projection mapping $P:W¥times F^{*}¥rightarrow W$.Definition 2. 2. Define$¥Omega_{¥phi}(x, f^{*})=P(¥Omega(x, f^{*}))$ , $D_{¥phi^{+}}(x, f^{*})=P(D^{+}(x, f^{*}))$ , and $J_{¥phi^{+}}(x, f^{*})=P(J^{+}(x$,

$f^{*}))$ . $¥Omega_{¥phi}(x, f^{*})$ is called the positive extended limit set, $D_{¥phi^{+}}(x, f^{*})$ the positiveextended profongationaf limit set of the solution $¥phi(x, f^{*}, t)$ .

Remark 2. 1. It is clear that for any $(x, f^{*})¥in W¥times F^{*}$, $D_{¥phi^{+}}(x, f^{*})=¥Upsilon_{¥phi^{+}}(x$,$f^{*})¥cup J_{¥phi^{+}}(x, f^{*})$ and $¥Omega_{¥phi}(x, f^{*})¥subset J_{¥phi^{+}}(x, f^{*})$ , where $¥gamma_{¥phi^{+}}(x, f^{*})=P(¥mathcal{T}^{+}(x, f^{*}))$ .

Remark 2. 2. If $(y, g)¥in J^{+}(x, f)$ , then $g¥in¥Omega_{f}$ . For, by definition of $J^{+}$

$(x, f)$ , there are sequences $f^{t}¥in F^{*}$ , $f^{t}¥rightarrow f$ and $ t^{t}¥rightarrow 4-¥infty$ such that $¥pi^{*}(f^{t}, t^{t})¥rightarrow g$.But we can assume that $f^{t}=¥pi^{*}(f, T^{t})$ for some sequence $¥{T^{t}¥}$ in 1, as $F$ is anopen set (recall the remark explained in Standing Hypothesis above). Obviously$T^{t}¥rightarrow 0$, otherwise it gives a contradiction to the fact that $f¥not¥subset¥Omega_{f^{*}}¥cup A_{f^{*}}$ . Thisimplies that $¥pi^{*}(¥pi^{*}(f, T^{t}), t^{t})=¥pi^{*}(f, T^{t}+t^{i})$ and $ T^{t}+t^{t}¥rightarrow¥infty$ . Hence $g¥in¥Omega_{f^{*}}$ .

It is proved in [7] that if the motion $¥pi_{f^{*}}$ in $F^{*}$ is positively compact, then$¥Omega_{¥phi}(x, f)$ can be characterized as follows.

Lemma 2. 1. [7, Lemma 2]. Let $f¥in C$ be a regular function and assume

102 S. M. Shamim IMDADI and M. Rama Mohana RAO

that the motion $¥pi_{f^{*}}$ is positively compact. Then a point $¥theta$ lies in $¥Omega_{¥phi}(x, f)$ ifand only if there is a sequence $¥{¥tau^{n}¥}$ in $R$ with $¥tau^{n}¥rightarrow+¥infty$ and $¥phi(x, f, ¥tau^{n})¥rightarrow¥$ $.

Similar characterizations for $J_{¥phi^{+}}(x, f)$ and $D_{¥phi^{+}}(x, f)$ are given in the fol-lowing two lemmas.

Lemma 2. 2. Let $f¥in C$ be a regular function and assume that the motion$¥pi_{f^{*}}$ is positively compact. Then a point $ff$ lies in $J_{¥phi^{+}}(x, f)$ if and only if $¥phi$

$(x^{n}, f, ¥tau^{n})¥rightarrow ¥mathrm{f}$ for some sequences $x^{n}¥rightarrow x$ and $¥tau^{n}¥rightarrow+¥infty$ .Proof. Let $¥hat{x}¥in J_{¥phi^{+}}(x, f)$ . By the definition of $J_{¥phi^{+}}(x, f)$ , there is an $f$ in

$F^{*}$ such that $(¥hat{x},¥hat{f})¥in J^{+}(x, f)$ . Further, there are sequences $¥{(y^{n}, g^{n})¥}$ and$¥{t^{n}¥}$ such that $(y^{n}, g^{n})¥rightarrow(x, f)$ , $ t^{n}¥rightarrow+¥infty$ , and $¥pi(y^{n}, g^{n}; ¥tau^{n})¥rightarrow(¥hat{x},¥hat{f})$ . This impliesthat there are sequences $¥{y^{n}¥}$ , $¥{g^{n}¥}$ and $¥{t^{n}¥}$ such that $y^{n}¥rightarrow x$, $g^{n}¥rightarrow f$, $ t^{n}¥rightarrow+¥infty$

and $¥phi(y^{n}, g^{n}, t^{n})¥rightarrow¥hat{x}$ . Now we can assume that the sequence $¥{g^{n}¥}$ is of the form$¥{¥pi^{*}(f, T^{n})¥}$ , where $T^{n}¥rightarrow 0$ (see Remark 2. 2). Let $¥phi(y^{n}, ¥pi^{*}(f, T^{n}), -T^{n})=x^{n}$.As $y^{n}¥rightarrow x$ and $T^{n}¥rightarrow 0$ , we have $x^{n}¥rightarrow x$ and $¥phi(x^{n}, f, t^{n}+T^{n})¥rightarrow¥hat{x}$ . Hence, if $¥phi¥in$

$J_{¥phi^{+}}(x, f)$ then there are sequences $x^{n}¥rightarrow x$ and $¥tau^{n}=t^{n}+T^{n}¥rightarrow¥infty$ such that $¥phi(x^{n}$ ,$f$, $¥tau^{n})¥rightarrow¥hat{x}$ . Conversely, suppose that $¥phi(x^{n}, f, ¥tau^{n})¥rightarrow¥theta$ for some sequences $x^{n}¥rightarrow x$

and $¥tau^{n}¥rightarrow+¥infty$ . Consider the sequence $¥{¥pi^{*}(f, ¥tau^{n})¥}$ . Since $¥pi_{f^{*}}$ is positively com-

pact, there exists a convergent subsequence $¥{¥pi^{*}(f, ¥tau^{n_{k}})¥}$ of $¥{¥pi^{*}(f, ¥tau^{n})¥}$ . Let $¥pi^{*}$

$(f, ¥tau^{n_{k}})¥rightarrow f¥in F^{*}$ . Thus, we have $x^{n_{k}}¥rightarrow x$ , $¥tau^{n_{h}}¥rightarrow+¥infty$ , $¥phi$ $(x^{n_{k}}, f, ¥tau^{n_{k}})¥rightarrow¥hat{x}$ and $¥pi^{*}$

$(f, ¥tau^{n_{k}})¥rightarrow¥hat{f}$ . This implies that $¥pi(x^{n_{k}}, f, ¥tau^{n_{k}})¥rightarrow(¥hat{x}, f)$ for some sequences $x^{n_{k}}¥rightarrow x$

and $¥tau^{n_{k}}¥rightarrow+¥infty$ . Therefore $(¥delta,¥hat{f})¥in J^{+}(x, f)$ , which implies that $¥&¥in J_{¥phi^{+}}(x, f)$ .This completes the proof.

Lemma 2. 3. Let $f¥in C$ be a regular function and assume that the motion$¥pi_{f^{*}}$ is positively compact. Then a point $¥hat{x}$ lies in $D_{¥phi^{+}}(x, f)$ if and only if $¥phi$

$(x^{n}, f, ¥tau^{n})¥rightarrow¥theta$ for some sequences $x^{n}¥rightarrow x$ , $¥tau^{n}¥geq 0$ .Proof. If $¥hat{x}¥in D_{¥phi^{+}}(x, f)$ , then, by Remark 2. 1, either $¥theta¥in¥gamma_{¥phi^{+}}(x, f)$ or $ x¥in$

$J_{¥phi^{+}}(x, f)$ . If $¥hat{x}¥in¥gamma_{¥phi^{+}}(x, f)$ , then there is a $¥tau¥geq 0$ such that $¥phi(x, f, ¥tau)=k$ andhence $¥phi(x^{n}, f, ¥tau^{n})¥rightarrow¥theta$ for the constant sequences $x^{n}=x$ and $¥tau^{n}=¥tau$. The rest ofthe proof is similar to that of Lemma 2. 2 and hence omitted.

3. Now we shall give some theorems which relate the compactness propertyof solutions of (2. 1) with the corresponding behavior of solutions of (2. 2).

Theorem 3. 1. Let $f¥in C$ be a regular function and assume that the mo-

tion $¥pi_{f^{*}}$ is compact. Let $x¥in W$ and $N$ be a neighborhood of $x$ such that Cl$[¥bigcup_{¥in R}¥{¥emptyset(y, f_{¥tau}, t) : y¥in N, t¥geq 0¥}]$ is a compact subset of W. Then the sets $D^{+}(x, f)$

and $J^{+}(x, f)$ are nonempty, compact and connected. If $(x^{*}, f^{*})¥in J^{+}(x, f)$ ,

then the solution $¥phi(x^{*}, f^{*}, t)$ of (2. 2) is compact. Furthermore, $D_{¥phi^{+}}(x, f)$ and$J_{¥phi^{+}}(x, f)$ are nonempty, compact and connected.

Proof. Let $N$ be a neighborhood of $x$ such that Cl [$¥bigcup_{¥tau¥in R}¥{¥phi(y, f_{¥tau}, t)$ : $y¥in N$, $t$

Prolongations and Prolongational Limit Sets 103

$¥geq 0¥}]$ be a compact subset of $W$. Consider the set $N¥times F$, where $F=¥{f_{f} : ¥tau¥in R¥}$ .Obviously $N¥times F$ is a neighborhood of $(x, f)$ . Further, we have Cl[$¥pi(N¥times F$ ;

$R^{+})]¥subset ¥mathrm{C}1[¥tau¥in R¥cup¥{¥emptyset(y, f_{r}, t) : y¥in N, t¥geq 0¥}]¥times F^{*}$, where $¥pi(N¥times F;R^{+})$ denotes the set$¥{¥pi(x, f_{¥tau} ;t) : (x, f_{¥tau})¥in N¥times F, t¥geq 0¥}$ . Since the motion $¥pi_{f^{*}}$ is compact, $F^{*}$ is com-

pact and hence $Cl[¥pi(N¥times F;R^{+})]$ is compact. Now the application of Theorem6. 7 [2, p. 63] yields that the sets $D^{+}(x, f)$ and $J^{+}(x, f)$ are nonempty, compactand connected. The rest of the proof follows from the fact that the projectionmapping $P:W¥times F^{*}Wi¥dot{s}$ continuous.

Remark 3. 1. G. R. Sell [7, Theorem 1] assumed that the solution $¥phi(x,$ $f$,$t)$ of (2. 1) is positively compact to prove a similar result for the $¥omega$ -limit set $¥Omega_{¥phi}$

$(x, f)$ . The following example shows that the assumption of positive compact-ness of the solution $¥phi(x, f, t)$ is not sufficient to conclude Theorem 3. 2.

Example 3. 1. Consider the differential system$i_{1}=-x_{1}/(t+1),¥dot{x}_{2}=x_{2}/(t+1)$ , $t¥geq 0$ .

It is easy to show that for any $x=(a, 0)¥in R^{2}$ , where $a¥neq 0$ , $¥phi(x, f, t)$ is positivelycompact but $J_{¥phi^{+}}(x, f)=¥{(x_{1}, x_{2}) : x_{1}=0, x_{2}¥in R¥}$ is not compact.

Theorem 3. 2. Let $f¥in C$ be a regular function and assume that the motion$¥pi_{f^{*}}$ is positively compact. Let $D^{+}(x, f)$ be a compact subset of $W¥times F^{*}$ . Thenfor every point $(x^{*}, f^{*})¥in J^{+}(x, f)$ , the solution $¥phi(x^{*}, f^{*}, t)$ of (2. 2) is cornpact.Moreover, there exist sequences $¥{x^{n}¥}$ in $W$ and $¥{¥tau^{n}¥}$ in $R$ with $x^{n}¥rightarrow x$, $¥tau^{n}¥rightarrow+¥infty$

such that $¥phi(x^{n}, f, ¥tau^{n}+t)$ converges to $¥phi(x^{*}, f^{*}, t)$ uniformly on compact sets in$R$ .

Proof. The first part follows from Theorem 3. 1. For the second part,since $¥phi(x^{*}, f^{*}, t)$ is compact it follows that $I_{(x^{*}.f^{*})}=R$ . Therefore, the solution$¥phi(x^{*}, f^{*}, t)$ and the motion $¥pi(x^{*}, f^{*}; t)$ are defined for all $t$ in $R$ . Now let $¥{x^{n}¥}$

and $¥{¥tau^{n}¥}$ be sequences in $W$ and $R$ respectively, with $x^{n}¥rightarrow x$, $¥tau^{n}¥rightarrow+¥infty$ and $¥pi(x^{n}$ ,$f;¥tau^{n})¥rightarrow(x^{*}, f^{*})$ . Since $¥pi$ is continuous $¥pi(x^{n}, f;¥tau^{n}+t)$ converges to $¥pi(x^{*}, f^{*}; t)$

uniformly on compact sets in $R$ [$6$ , Lemma 4]. Further, since the projectionmapping $P$ is continuous, the solutions $¥phi(x^{n}, f, ¥tau^{n}+t)=P(¥pi(x^{n}, f, ¥tau^{n}+t))$ con-verge to $¥phi(x^{*}, f^{*}, t)$ uniformly on compact sets in $R$ . This completes theproof.

The conclusion of Theorem 3. 2 can be formulated in terms of the positiveprolongational limit set $J_{¥phi^{+}}(x, f)$ . The basic fact here is that if $x^{*}¥in J_{¥phi^{+}}(x, f)$ ,then there is an $f^{*}$ in $F^{*}$ such that $(x^{*}, f^{*})¥in J^{+}(x, f)$ .

Corollary 3. 1. Let $f$, $¥pi_{f^{*}}$ , and $D^{+}(x, f)$ be as in the preceding theorem.Then for every point $x^{*}¥in J_{¥phi^{+}}(x, f)$ , there is a function $f^{*}$ in $¥Omega_{f^{*}}$ such thatthe solution $¥phi(x^{*}, f^{*}, t)$ of (2. 2) is compact.

This corollary asserts that the positive prolongational limit set $J_{¥phi^{+}}(x, f)$ isquasi-invariant in the sense defined by R. K. Miller [5].

104 S. M. Shamim IMDADI and M. Rama Mohana RAO

If $f$ is asymptotically autonomous [7], that is, if $¥Omega_{f^{*}}$ is a singleton, thenone can say more.

Corollary 3. 2. Let $f¥in C$ be a regular function that is asymptoticallyautonomous. Let $¥Omega_{f^{*}}=¥{f^{*}¥}$ . Then every positive prolongational limit set $J^{+}$

$(x, f)$ can be expressed in the form$J^{+}(x, f)=J_{¥phi^{+}}(x, f)¥times¥{f^{*}¥}$ .

Therefore, $J_{¥phi^{+}}(x, f)$ is the union of solutions of $x^{f}=f^{*}(x)$ . If $D^{+}(x, f)$ iscompact, then for every $x^{*}$ in $J_{¥phi^{+}}(x, f)$ , the solution $¥phi(x^{*}, f^{*}, t)$ is compact.

Theorem 3. 3. Let $f¥in C$ be regular function and assume that the motion$¥pi_{f^{*}}$ is positively compact. Let $¥Omega_{¥phi}(x, f)$ be a nonempty compact subset of $W$.Then the solution $¥phi(x, f, t)$ of (2. 1) is positively compact.

Proof. Consider the set $¥Omega(x, f)$ . Since $¥Omega(x, f)¥subset¥Omega_{¥phi}(x, f)¥times F^{*}$ and $¥Omega(x$,

$f)$ is closed, obviously the set $¥Omega(x, f)$ is compact. This implies that the clo-sure of $¥gamma+(x, f)$ is compact [2, Lemma 5. 8]. Further, since the projectionmapping $P$ is continuous, it follows that the solution $¥phi(x, f, t)$ of (2. 1) is po-

sitively compact.

Theorem 3. 4. Let $f¥in C$ be a regular function and assume that the motion$¥pi_{f^{*}}$ is positively compact. Then $¥Omega_{¥phi}(x, f)$ is nonempty whenever $J_{¥phi^{+}}(x, f)$ isnonempty and compact.

The proof is direct and simple and hence omitted.

4. In this section we shall give some results which characterize the stability

property of solutions of (2. 1) in terms of its prolongation and prolongationallimit sets.

Lemma 4. 1. Let $f¥in C$ be a regular function and assume that the motion$¥pi_{f^{*}}$ is positively compact. If the solution $¥phi(x, f, s)$ of (2. 1) is stable, then

$D_{¥phi^{+}}(x, f)=$ Cl $[¥gamma_{¥phi^{+}}(x, f)]$ .

Proof. We know that $D_{¥phi^{+}}(x, f)=¥gamma_{¥phi^{+}}(x, f)$ $¥cup J_{¥phi^{+}}(x, f)$ and Cl $[¥gamma_{¥phi^{+}}(x, f)]$

$=r_{¥phi^{+}}(x, f)¥cup¥Omega_{¥phi}(x, f)$ . Therefore, we only need to prove that $J_{¥phi^{+}}(x, f)=¥Omega_{¥phi}(x$,

$f)$ . Let $y¥in J_{¥phi^{+}}(x, f)$ . Then, by Lemma 2. 2, there exist sequences $x^{n}¥rightarrow x$ and$¥tau^{n}¥rightarrow+¥infty$ such that $¥phi$ $(x^{n}, f, ¥tau^{n})¥rightarrow y$ . Let $¥epsilon>0$ be given. Since $¥phi(x, f, t)$ is sta-$¥mathrm{b}¥mathrm{l}¥mathrm{e}$ , for given $¥epsilon/2$ there exists a $¥delta=¥delta(¥epsilon/2)>0$ such that

$|¥phi(x, f, s)$ $-¥phi(z, f, t)|<¥epsilon/2$

for all $t¥geq 0$ , whenever $|x-z|<¥delta$ . As $x^{n}¥rightarrow x$ there is an $N_{1}¥geq 0$ such that $|x-x^{n}|$

$<¥delta$ for $n¥geq N_{1}$ . Thus, we have(4. 1) $|¥phi(x, f, ¥tau^{n})-¥phi(x^{n}, f, ¥tau^{n})|<¥epsilon/2$ for $n¥geq N_{1}$ .

Further, since $¥phi(x^{n}, f, ¥tau^{n})¥rightarrow y$ , given $¥epsilon/2$ there exists a number $N_{2}¥geq 0$ such that(4. 2) $|¥phi(x^{n}, f, ¥tau^{n})-y|<¥mathrm{e}/2$ for $n¥geq N_{2}$.

Choose $N=¥max(N_{1}, N_{2})$ . From (4. 1) and (4. 2), we have

Prolongations and Prolongational $Liw¥iota it$ Sets 105

$|¥phi(x, f, ¥tau^{n})-y|¥leq|¥phi(x, f, ¥tau^{n})-¥phi(x^{n}, f, ¥tau^{n})|+|¥phi(x^{n}, f, ¥tau^{n})-y|<¥epsilon/2+¥epsilon/2$

$=¥epsilon$ for $n¥geq N$.This shows that $¥phi(x, f, ¥tau^{n})¥rightarrow y$ . Therefore $y¥in¥Omega_{¥phi}(x, f)$ . Hence $D_{¥phi^{+}}(x, f)=¥mathrm{C}1$

$[¥mathcal{T}_{¥phi^{+}}(x, f)]$ . This completes the proof.Theorem 4. 1. Let $f¥in C$ be a regular function with $f(0, t)=0$ for all $ t¥geq$

$0$ and assume that the motion $¥pi_{f^{*}}$ is positively compact. Then the null solution

of (2. 1) is stable if and only if $D_{¥phi^{+}}(0, f)=¥{0¥}$ .The necessity part of the proof follows from Lemma 4. 1 and sufficiency

part is direct and hence omitted.Corollary 4. 1. Let $f¥in C$ be a regular function with $f(0, t)=0$ for $afl$ $ t¥geq$

$0$ and assume that the motion $¥pi_{f^{*}}$ is positively compact. Then the null solution

of (2. 1) is stable if the local dynamical system $¥pi$ on $W¥times F^{*}$ is stable.Proof. The stability of $¥pi$ implies that for each $(x, f)¥in W¥times F^{*}$ , $D^{+}(x, f)$

$=$ Cl $[¥gamma+(x, f)]$ [$2$ , Def. 7. 7 and Theorem 7. 8]. Therefore $D^{+}(0, f)=¥mathrm{C}1[¥mathcal{T}^{+}(0$,$f)]$ and by taking the projection $P$ of both the sets on $W$, we have $D_{¥phi^{+}}(0, f)$

$=¥{0¥}$ . Thus the application of Theorem 4. 1 yields the desired result.Theorem 4. 2. Let $f¥in C$ be a regular function with $f(0, t)=0t¥geq 0$ and

assume that the motion $¥pi_{f^{*}}$ is positively compact. Then the null solution of(2. 1) is asymptoticaffy stable if and only if $¥{x_{0} : J_{¥phi^{+}}(x_{0}, f)=¥{0¥}¥}$ is a neighbor-hood of the origin.

Proof. Let the null solution of (2. 1) be asymptotically stable. Hence itis stable. Therefore, by Theorem 4. $¥mathrm{W}$ , we have $D_{¥phi^{+}}(0, f)=¥{0¥}$ . Further, fora given $¥delta_{0}>0$ and for each $¥eta>0$ there exists a $T=T(¥eta)>0$ such that $|x_{0}|<¥delta_{0}$

implies that $|¥phi(x_{0}, f, t)|<¥eta$ for all $t$ $>T$. This implies that $J_{¥phi^{+}}(x_{0}, f)=¥{0¥}$ ,

because by taking sequences $x^{n}¥rightarrow x_{0}$ and $¥tau^{n}¥rightarrow+¥infty$ , one can easily prove that $¥phi$

$(x^{n}, f, ¥tau^{n})¥rightarrow 0$ . Hence $¥{x_{0} : |x_{0}|<¥delta_{0}¥}$ is the neighborhood of the origin such that$J_{¥phi^{+}}(x_{0}, f)=¥{0¥}$ . Conversely, suppose that $¥{x_{0} : J_{¥phi^{+}}(x_{0}, f)=¥{0¥}¥}$ is a neighbor-hood of the origin. This shows that $J_{¥phi^{+}}(0, f)=¥{0¥}$ . Further, we have $D_{¥phi^{+}}(0$ ,$f)=¥gamma_{¥phi^{+}}(0, f)¥cup J_{¥phi^{+}}(0, f)=¥{0¥}$ . Hence by Theorem 4. 1, it follows that the nullsolution of (2. 1) is stable. For the rest of the proof, let $¥{x_{0} : |x_{0}|<¥delta_{0}¥}¥subset¥{x_{0}$ :$J_{¥phi^{+}}(x_{0}, f)=¥{0¥}¥}$ . Then since for each $x_{0}$ satisfying $|x_{0}|<¥delta_{0},J_{¥phi^{+}}(x_{0}, f)$ is a no-

nempty compact set, we have by Theorem 3. 4 that the set $¥Omega_{¥phi}(x_{0}, f)$ is nonem-

pty. Therefore $¥Omega_{¥phi}(x_{0}, f)=¥{0¥}$ whenever $|x_{0}|<¥delta_{0}$ . This implies that for a given$¥eta>0$ there exists a $T=T(¥eta)$ such that whenever $|x_{0}|¥backslash ^{/}¥delta_{0}$ , $|¥phi(x_{0}, f, t)|<¥eta$ for all$t¥geq T$. This completes the proof.

The authors wish to thank the referee for his valuable suggestions.

106 S. M. Shamim IMDADI and M. Rama Mohana RAO

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and saddle sets, Contrib. Differential Equations 3 (1963), 249-294.nuna adreso:Department of MathematicsIndian Institute of TechnologyKanpur-208016, India

(Ricevita la 3-an de Decembro, 1973)(Reviziita la 25-an de Oktobro, 1974)